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Snakes With an Ellipse-Reproducing Property
Ricard Delgado-Gonzalo, Philippe Thévenaz, Chandra Sekhar Seelamantula, Member, IEEE, and
Michael Unser, Fellow, IEEE
Abstract—We present a new class of continuously defined parametric snakes using a special kind of exponential splines as basis
functions. We have enforced our bases to have the shortest possible
support subject to some design constraints to maximize efficiency.
While the resulting snakes are versatile enough to provide a good
approximation of any closed curve in the plane, their most important feature is the fact that they admit ellipses within their span.
Thus, they can perfectly generate circular and elliptical shapes.
These features are appropriate to delineate cross sections of cylindrical-like conduits and to outline bloblike objects. We address the
implementation details and illustrate the capabilities of our snake
with synthetic and real data.
Index Terms—Active contour, exponential B-spline, parameterization, parametric snake, segmentation.
I. INTRODUCTION
CTIVE contours and, in particular, snakes are effective
tools for image segmentation. Within an image, an active
contour is a curve that evolves from an initial position, which
is usually specified by a user, toward the boundary of an object. The evolution of the curve is formulated as a minimization
problem. The associated cost function is called snake energy.
Snakes have become popular because it is possible for the user
to interact with them not only when specifying its initial position but also during the segmentation process.
Research in this area has been fruitful and has resulted in
many snake variants [1], [2]. They differ in the type of curve
representation and in the choice of the energy term [3]. Snakes
can be broadly categorized in terms of curve representation as
follows:
1) point-snakes, where the curve is described in a discrete
fashion by a set of points [4]–[6];
2) parametric snakes, where the curve is described continuously by some coefficients using basis functions [7]–[11];
3) implicit snakes, where the representation of the curve is implicit and described as the level set of a surface [12]–[15].
Point-snakes can be viewed as a special case of parametric
snakes where a large number of coefficients is used [10].
A
Manuscript received March 23, 2011; revised July 20, 2011 and September
12, 2011; accepted September 19, 2011. Date of publication September 29,
2011; date of current version February 17, 2012. This work was supported in
part by Swiss SystemsX.ch under Grant 2008/005 and in part by the Swiss National Science Foundation under Grant 200020-121763. The associate editor
coordinating the review of this manuscript and approving it for publication was
Prof. Ferran Marques.
R. Delgado-Gonzalo, P. Thévenaz, and M. Unser are with the Biomedical
Imaging Group, École polytechnique fédérale de Lausanne, 1015 Lausanne,
Switzerland.
C. S. Seelamantula is with the Department of Electrical Engineering, Indian
Institute of Science, Bangalore 560 012, India.
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TIP.2011.2169975
Fig. 1. Approximation capabilities of the proposed parametric snake. The thin
solid line corresponds to an elliptical fit. The dashed thick line corresponds to a
generalized shape.
Parametric snakes require fewer parameters and result in faster
optimization. It can be shown that the computation complexity
of the snake energy and, therefore, the speed of the optimization algorithms is related to the size of the support of the basis
functions [3]. It is therefore critical to minimize this support
while designing parametric snakes. The curve of parametric
snakes is represented explicitly. It is then easy to introduce
smoothness and shape constraints [7]. It is also straightforward
to accommodate user interaction. This is often achieved by
allowing the user to specify some anchor points that the curve
should go through [4]. The downsize of the method is that the
topology of the curve is imposed by the parameterization. This
makes parametric snakes less suitable for handling topological
changes, although solutions have been proposed for specific
cases [16], [17].
Implicit approaches offer great flexibility as far as the curve
topology is considered [18]. However, they tend to be computationally more expensive since they evolve a 2-D surface rather
than a 1-D curve.
In this paper, we design fast parametric snakes capable of perfectly outlining elliptic objects and yet versatile enough to provide a close approximation of any closed curve in the plane.
We illustrate in Fig. 1 how our snake can adopt the shape of
a perfect ellipse (i.e., reproduces the ellipse) as well as more
refined shapes. Segmenting circles and ellipses in images is a
problem that arises in many fields, such as biomedical engineering [19]–[22] or computer graphics [23], [24]. In medical
imaging in particular, it is usually necessary to segment arteries
and veins within tomographic slices [25]. Because those objects are physiological tubes, their section show up as ellipses
in the image. Ellipse-like objects are also present at microscopic
scales. For instance, cell nuclei are known to be nearly circular
1057-7149/$26.00 © 2011 IEEE
DELGADO-GONZALO et al.: SNAKES WITH ELLIPSE-REPRODUCING PROPERTY
[26]. Water drops are similarly spherical due to surface tension
forces [27]. However, these elements deform and become elliptical when they are subject to stress forces.
In order to segment efficiently elliptical objects, a parametric
snake called the Ovuscule was proposed in [28]. It is a minimalistic elliptical snake defined by three control points. Its main
drawback was that it was unable to represent shapes different
from circles and ellipses. Our goal here is to create a more versatile parametric snake whose basis functions are short, reproduce
ellipses perfectly, and have good approximation properties. Our
main contribution in this paper is to fulfill this goal by selecting
a special kind of exponential B-splines. We are able to prove
that our basis functions are the ones with the shortest support
among all admissible functions. Since the computational cost
of spline snakes is determined in part by the size of the support
of the basis function, our use of the shortest possible support favors optimal performance.
This paper is organized as follows. In Section II, we review
the general parametric snake model and formalize our design
constraints. Our main contribution is described in Section III,
where we build an explicit expression for the underlying basis
functions that fulfill our requirements, and we analyze in detail
its reproduction and approximation properties. Implementation
details such as energy functionals and discretization issues are
addressed in Section IV. Finally, we perform report evaluations
in Section V.
II. PARAMETRIC SNAKES
A. Parametric Representation of Closed Curves
on the plane can be described by a pair of CarteCurve
sian coordinate functions
and
, where
is a
and
are efficontinuous parameter. The 1-D functions
ciently parameterized by linear combinations of suitable basis
functions. Among all possible bases, we focus on those derived
from the compactly supported generator and its integer shifts
. This allows us to take advantage of the availability of fast and stable interpolation algorithms [29].
We are interested in close curves specified by an -periodic
sequence of control points
, with
.
The parametric representation of the curve is then given by the
following vectorial equation:
(1)
The number of control points
determines the degrees of
freedom in model (1). Small numbers lead to constrained shapes
and large numbers lead to additional flexibility and more general
shapes.
Since curve is closed, each coordinate function is periodic,
and the period is common for both. For simplicity, in (1), we
normalized this period to be unity. Under these conditions, we
can reduce the infinite summation in (1) to a finite one involving
periodized basis functions as
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(2)
is the -periodization of the basis function .
where
This kind of curve parameterization is general. Using this
model, we can approximate any closed curve as accurately as
desired by using a higher number of vector coefficients
, under the condition satisfies some mild conditions [30].
B. Desirable Properties for the Basis Functions
We now enumerate the conditions that our parametric snake
model should satisfy and introduce the corresponding mathematical formalism.
1) Unique and Stable Representation. We want our parametric curve to be defined in terms of the coefficients in
such a way that unicity of representation is satisfied. Furthermore, for computational purposes, we ask the interpolation procedure to be numerically stable.
The generating function is said to satisfy the Riesz basis
condition if and only if there exist two constants
such that
(3)
for all
. A direct consequence of the lower inequality
for all
is that condition
implies that
for all
. Thus, the basis
functions are linearly independent, and every function is
uniquely specified by its coefficients. The upper inequality
ensures the stability of the interpolation process [29].
It has been shown in [31] that, due to the integer-shift-invariant structure of the representation, the Riesz condition
has the following equivalent expression in the Fourier domain:
where
denotes the Fourier
transform of . Once expressed in the Fourier domain, the
Riesz condition provides a practical way to verify if a given
generating function satisfies (3).
2) Affine Invariance. Since we are interested in outlining
shapes irrespective of their position and orientation, we
would like our model to be invariant to affine transformations, which we formalize as
(4)
where is a (2 2) matrix and is a 2-D vector. Using
(4), it is easy to show that affine invariance is ensured if
and only if
(5)
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Theorem 1: The centered generating function with minimal
support that satisfies all conditions in Section II-B and reprocoefficients is
duces sinusoids of unit period with
(8)
M = 10
Fig. 2. (a) Parametric representation of the unit circle and (b) its coordinate
. The dashed lines in (b)
functions with exponential B-splines and
indicate the corresponding basis functions.
In the literature, this constraint is often called the partition-of-unity condition [29].
3) Well-Defined Curvature. The curvature of a parametric
is given by
curve at point
In order to prove Theorem 1, we refer to the distributional
decomposition theorem detailed in [32]. This decomposition
theorem provides a complete characterization of the family of
basis functions with minimum support that reproduce exponential polynomials. It states that every minimum-support function
that reproduces exponentials
, for all
with
, can be written as
(9)
where is an arbitrary shift parameter that corresponds to the
is the approlower extremity of the support of and where
priate exponential B-spline defined as follows:
e
j
where the dot denotes the derivative with respect to . We
would like to be able to compute for every point on the
snake. To do so, each coordinate function (or, equivalently,
basis ) must be at least twice differentiable, and its second
derivative must be bounded.
III. REPRODUCTION OF ELLIPSES
Since every ellipse can be obtained by applying an affine
transformation to the unit circle, we focus on the reproduction
of this simpler shape. This simplification is allowed whenever
the affine-invariance requirement stated in Section II-B is satisfied.
vectorial coefficients
A parametric snake defined by
and by a generating function is said to reproduce the unit
and
circle if there exist two -periodic sequences
such that
(6)
(7)
That is, we need to be able to reproduce sinusoids of unit period for each component of the parametric snake, as illustrated
in Fig. 2. Note that, when (6) and (7) hold, it is possible to represent any sinusoid of unit period for an arbitrary initial phase
using linear combinations of the two sequences of coefficients.
A. Minimum-Support Ellipse-Reproducing Basis
We now present and prove our main result. We provide an explicit expression for the minimum-support basis functions that
reproduce sinusoids.
(10)
Note that exponential B-splines are entirely specified by collection
. The ordering of the poles
is
irrelevant. A complete survey of the properties of exponential
B-splines can be found in [33].
We finally have the mathematical tools to justify our choice
for the generating function in (8).
Proof: Using (9), we see that needs to be constructed
from combinations of exponential B-splines with parameters
and
. Therefore, we
have
(11)
This ensures that is the shortest generating function that
reproduces constants and all sinusoids of unit period with
coefficients. The constant-reproduction property is a direct con, and the sinusoid-reproduction propsequence of using
,
,
erty can be proved by using
and Euler’s identity.
Using the properties of exponential B-splines, we know that
is twice differentiable. Moreover, the second derivative is
bounded but may be discontinuous. Therefore, and in (11)
must vanish to ensure that the curvature of the snake is well
can be computed by
defined. Since reproduces constants,
imposing the partition-of-unity condition. From (5), we have
Exponential B-splines parameterized by form a Riesz basis
for all pairs such that
if and only if
. In our case, it is important to realize that this condition is
satisfied if and only if
. In other words, at least three
control points are needed to define our parametric snake.
DELGADO-GONZALO et al.: SNAKES WITH ELLIPSE-REPRODUCING PROPERTY
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where
is the best approximation within the span
. Since is usually unknown, we measure the error
averaged over all possible shifts as
(12)
, folWe give in Section III-C the decay of as
lowing the method described in [30]. As expected, we find that
when
the best averaged-quadratic-mean error decays as
increases, which results in
the number of vector coefficients
the same rate as the quadratic B-spline [34].
Fig. 3. Plot of the quadratic B-spline and the resulting generating functions
, 4, 5, and 6. The function with the lowest peak at t
given in (8) for M
corresponds to M
, and as M increases, the height of the central peak
increases as well.
=3
=3
=0
Finally, a closed form for is obtained by applying the inverse Fourier transform to (11), which yields
e
e
j
e
j
e
j
j
j
where we have set
in order to ensure that the basis
function is centered.
We show in Fig. 3 some members of this family of functions
for several values of . We observe that they share with the
. Likewise,
quadratic B-spline a finite support of length
they are one-time continuously differentiable and have a similar
bumplike appearance.
B. Approximation Properties of
We are not only interested in reproducing ellipses but would
also like our snake to be able to approximate any other shape .
This is achieved by increasing the number of degrees of freedom
of nodes. In the Fourier domain, it
afforded by the number
inis easy to see that converges to a quadratic B-spline as
creases. Therefore, we expect similar approximation properties
for large values of .
While leads to integer-shift invariance, the space spanned
by the generating function is not shift invariant in general.
vector coefficients is
Hence, the approximation error using
dependent upon a shift in the continuous parameter of the function of unit period . The minimum-mean-square approximation
error for a shifted function is given by
C. Approximation Order of
In this section, we introduce the necessary formalism to compute the order of the approximation error associated to the best
possible approximation of the periodic vector function within
, where is given by (8).
the span of basis
As explained in Section III-B about the approximation properties of , the space spanned by the generating function is not
shift invariant in general. Hence, as a metric of dissimilarity between shapes, we use the averaged minimum-mean-square approximation error .
Using the main result of [30], we obtain the asymptotic behavior of as
where
, and
is the
th derivative of the Fourier transform of . Following lengthy
calculations, we get (13) and (14), shown at the bottom of the
. It can be shown
page, where we defined
that
and
. Since curve
does not depend on , we can also write that
which shows that the averaged quadratic mean error decays as
.
D. Best Constant and Ellipse Fitting
Since our snakes have the capability of perfectly reproducing
ellipses, it is natural to ask which is the best ellipse that approximates the parametric curve defined by the -periodic se. In other words, we are interested in finding
quence
ellipse that minimizes
(13)
(14)
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Since is continuous and has a unit period, we can expand it
in a Fourier series as
e
, we see that
Setting
complex exponential
, which is
convolve both sides of (18) with
reproduces the
-periodic. If we now
, we get that
(15)
e
The Fourier series vector coefficients
in (15) are given by
e
e
e
(16)
where the parametric expression of has been used in the second
equality.
From the classical theory of harmonic analysis, we know that
the best ellipse approximation (componentwise sinusoids) of ,
sense, is the first-order truncation of the series
in the
,
, and
are kept.
(15), where only the terms
Therefore, we have
where we have used the definition of from (8), along with
the fact that the convolution operator commutes with the shift
operator. To simplify the left-hand side, we invoke an important property of linear shift-invariant (LSI) systems: complex
exponentials are eigenfunctions of LSI operators. By virtue of
is presented at the
this property, if the complex exponential
input of a system specified by the impulse response , then its
, where denotes the Fourier transoutput is given by
as the impulse response
form of . If we consider
of an LSI system, then
e
(17)
where
is the center of gravity of the snake. The Fourier
coefficients in (17) can be obtained easily from (16) as
e
Therefore, we have that
e
e
By flipping the sign of we can easily obtain an analogous
result for the reproduction of
. Finally, by using both
results, we have
j
(19)
where
(20)
where
Since all sinusoids of unit period can be reproduced by the
generating function and the appropriate -periodic sequence
belongs to the span of . For the
of coefficients , curve
sake of completeness, we provide in the next section an explicit
expansion of sinusoids in terms of .
E. Expansion of Sinusoids With
Here, we explicitly find the sequence of
vector coefficients that reproduce sinusoids of unit period using the generating function given in (8). We start by recalling the exponential-reproducing property of the exponential B-splines as
e
e
(18)
Note that sequences and are -periodic and that the summations in (19) and (20) can be reduced to finite ones if we make
use of the periodized basis functions.
We have expressed in (19) and (20) how to compute the vector
coefficients for reproducing sinusoids of unit period and the ini. The appropriate linear combination of
tial phase of
and then allows one to reproduce sinusoids of arbitrary shape.
DELGADO-GONZALO et al.: SNAKES WITH ELLIPSE-REPRODUCING PROPERTY
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IV. IMPLEMENTATION
Since the presented parametric active contour is a spline
snake, it is capable of handling all traditional energies applicable to point and parametric snakes. However, to illustrate the
behavior of our parameterization in a real implementation, we
performed our experiments with a specific snake energy that
we designed to be versatile.
In this section, we first introduce the snake energy that drives
the optimization process, and then, we provide a description
of the implementation details for the proposed snake. We construct the energy functional to detect dark objects on a brighter
background.
Fig. 4. Schematic representation of a parametric snake r (dashed line), of its
interaction with an object constituted by a gray semicircle (representing low
pixel values), of the vector dx tangent to the curve, and of the gradient vector
f of the image. The vector k, which is mentioned in the text, is perpendicular
to the image plane and points toward the reader.
r
A. Snake Energy
The active-contour algorithm is always driven by a chosen
energy function. Thus, the quality of the segmentation depends
on the choice of this energy term. There are many construction strategies that can be categorized in two main families: 1)
edge-based schemes, which use gradient information to detect
contours [4], [7], [10]; and 2) region-based methods, which use
statistical information to distinguish different homogeneous regions [9], [35]. In order to benefit from the advantages of both
strategies, a unified energy was proposed in [3]. In our case, we
are going to follow a similar approach by using a convex combination of gradient energy and region energy, such as in
(21)
. The tradeoff parameter balances the conwhere
tribution of the edge-based energy and the region-based energy. Its value depends on the characteristics of each particular
application.
For the gradient-based (or edge) energy, we consider the one
described in [35] since it has the advantage of penalizing the
snake when the orientation is inconsistent with the object to segment. Let be our parametric snake. The contour energy term
is then given by
where
denotes the tangent vector of the curve in
the 3-D space formed by the image plane and its ordenotes the outward
thogonal dimension,
vector orthonormal to the image plane,
is the within-plane
gradient of image at
on the curve, and is the 3-D
cross product. In Fig. 4, we present the configuration of the
various quantities involved. The chirality of the system of coordinates will determine the sign of the integrand, as discussed
in [3] and [35]. Using Green’s theorem, the edge energy can be
also expressed as the surface integral
For the region-based energy, we adopt a strategy similar to
[28]. More precisely, our region-based energy discriminates an
around the
object from its background by building ellipse
snake and by maximizing the contrast between the intensity of
the data averaged within the curve and the intensity of the data
, the region
averaged over the elliptical shell . When
energy term can be expressed as
(23)
where
is given by
The normalization factor
can be interpreted as the signed
. The sign of quanarea, which is defined as
depends on the clockwise or the anticlockwise path foltity
lowed on curve . In this paper, we follow the usual convention
whereby an anticlockwise path leads to a positive sign. We enwhen takes
force our criterion to remain neutral
a constant value, such as in flat regions of the image. To achieve
.
this, we set
The construction of the elliptic shell is performed by using
the best ellipse given in (17) and by magnifying its axes by
factor to achieve
j
where
by curve
, and
is the signed area enclosed
, with
(22)
,
is the Laplacian of the image , and
where
is the region enclosed by .
In Fig. 5, we illustrate how we take advantage of the ideas
presented in Section III-D to build the best ellipse approximaof an arbitrary snake . Using constraint
,
tion
of the enclosing shell
.
we can determine the contour
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Fig. 6. Mean time of one iteration in the snake evolution.
Fig. 5. Representation of the parametric snake r, the best ellipse approximation
.
r , and the corresponding enclosing shell r used in E
D. Optimization
B. Accelerated Implementation
The computational cost is dominated by the evaluation of the
surface integrals in (22) and (23). An efficient way to implement
these operations is the use of preintegrated images. Let be the
, respectively) and
function that we are integrating ( , , or
). Then, by Green’s
let be the domain of integration ( or
theorem, we rewrite the surface integrals as the line integrals
where
is the boundary of
were able to determine this residual analytically, but in the interest of space, we do not provide it here.
and
(24)
(25)
The use of Green’s theorem to rewrite the surface integrals as
line integrals reduces dramatically the computational load. This
can only be achieved if the curve is defined continuously, such
as with the curves of Section II-A. By contrast, this acceleration would not be available to methods such as point snakes and
level sets because their implementation ultimately relies on discretization.
The optimization of the snake can be efficiently carried out
by Powell-like line-search methods [36]. These methods require
the derivatives of the energy function with respect to the parameters (i.e., the knot coefficients) and converge quadratically to
the solution. The algorithm proceeds as follows. First, one direction within the parameter space is chosen depending on the
partial derivatives of the energy. Second, 1-D minimization is
performed within the selected direction. Finally, a new direction is chosen using the partial derivatives of the energy function
once more while enforcing conjugation properties. This scheme
is repeated until convergence. Assuming a bilinear interpolation of the original function , we were able to derive exact and
closed expressions for these derivatives that take the residual of
the lookup table into account.
For spline snakes, it has been shown that the evaluation of the
partial derivatives of the energy of form (21) depends quadratically on the number of parameters [3]. In Fig. 6, we compare
the computational cost of the snake during line minimization
(simple update) and when the energy gradient is required to
chose a new direction (gradient update). For the latter case, we
contrast the computation time of an analytical computation of
the gradient to that of a centered finite-difference approach. For
low values of , the simple update and the gradient update
using analytical energy gradient lead to a similar computational
increases, the quadratic behavior of
load. As the value of
the computation of the gradient makes the update cost increase.
This quadratic behavior can be easily discerned in the topmost
curve of Fig. 6.
V. EXPERIMENTS
C. Sampling
Despite the fact that we are assuming a continuously defined
model for our functions, in a real-world implementation we only
have at our disposal a sampled version of the functions we want
to preintegrate. To solve this inconsistency, we perform a bilinear interpolation of the sampled data, and we store in lookup
tables the values of (24) and (25) at integer locations. Then, the
energies are obtained using the first approximation given by the
lookup tables. In our implementation, we have corrected them
by supplying a residual that allows us to get the exact result. We
We present in this section four experimental setups. In the first
one, we compare our choice in (8) against the classical quadratic
B-spline when representing sinusoids. We move away from sinusoids in the second experiment, where we work with synthetic
data and perform an objective validation of the segmentation
properties of our snake in noiseless and noisy environments. In
the fourth setup, we also perform a quantitative evaluation by
segmenting real cardiac MRI data. Finally, in the last experiment, we illustrate some real applications of our snake where
the ground truth is not available.
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Fig. 7. Approximations of a sine function with unit period. (a) Parametric representation (solid line) using ' (dashed lines). (b) Best parametric approximation (solid line) using (dashed lines).
Fig. 8. Sinusoid of period 3, its representation with our basis function (solid
line), and its best quadratic B-spline approximation (dashed line).
A. Approximation of Sinusoids
B. Accuracy and Robustness to Noise
By design, our basis function has the property of reproducing sinusoids exactly. By contrast, the classical polynomial
B-splines do not enjoy this property. In this section, we are focusing on this aspect and exhibit the amount of error committed
by B-splines when attempting to reproduce a sine function.
We start with the exact reproduction by our basis. Using the
result of Section III-E, we determine the coefficients for case
(smallest possible ). They are given by
In this section, two experiments are carried out. The first one
consists of outlining different synthetic bloblike shapes in a
noise-free environment. The second experiment consists of outlining one specific target within an image in the presence of
, i.e., we make use
noise. In both experiments, we set
of the region energy only. This particular choice ensures that
the snake is not mislead by noisy boundaries in the presence of
excessive of noise.
In the first experiment, we generate ten test images of size 512
512 by pixelwise sampling of our shape of interest, which
is built by intersecting or by making the union of two circles
with a radius of 50 pixel units. We illustrate these shapes in the
header of Table I. They are parameterized with distance , in
pixel units, between the centers of the circles. For
, the
,
shape is built by the intersection of the two circles. For
they are parameterized by their union. The grayscale values of
the images are 255 for the shape and 0 for the background.
We used the Jaccard distance
to measure as a percentage the dissimilarity between the two
corresponds to the ground-truth region, and
sets. There,
corresponds to the region enclosed by the snake. We computed
with a pixelwise discretization of the images.
In the simulations of Table I, we investigated the dependence
of coefficients and distance between
of on the number
when the snake did not
the circles. We denoted with a dash
converge; and, we could not compute the Jaccard distance. We
initialized every snake as a circle with a radius of 75 pixel and
a center that lay in the middle of the shape. We observe that
of conthe results in Table I tend to improve as the number
trol points is increased, particularly for the nonelliptical shapes.
However, the increase in the number of control points does not
bring any further improvement when the shape to segment is a
perfect circle. This result is expected since the circular shape is
. The residual error shown
reproduced exactly for any
can be attributed to the discretization of
in Table I for
and . We also observe that, for
and
,
and
the Jaccard distance starts increasing severely for
for
, respectively. This is due to the fact that the sharp
corners of the shape lead to loops in the curve during the optimization process. Such self-intersections violate the conditions
of Green’s theorem in Section IV-A.
where
corresponds to the periodization of the basis function
(8), as in (2).
We continue with approximate reproduction by B-splines.
For fairness, we choose the quadratic B-spline so that the size
of the support of
and is the same. The reproduction will be
approximate. This will happen not because of the limited size
of the support but because of the fact that the sine function does
not lie in the span of polynomial B-splines of any degree. Nevertheless, we can compute the coefficients that best adjust the
sinusoid with unit period in the least-squares sense. This yields
where
(26)
is the quadratic B-spline and the subscript 3 indicates a threeperiodized basis function as in (2).
We observe in Fig. 7 that both constructions result in sinelike functions. However, the reproduction is exact in the left
part of Fig. 7, whereas it is only approximate in the right part.
is idenThis happens, despite the fact that the support of
tical to the support of , the asymptotic approximation properties of
and are identical, and
and have the same degree of differentiability. We show in Fig. 8 the amount of error
committed by the parabolic approximation. We determine that
.
MSE
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TABLE I
ERROR PERCENTAGE OF OUR SNAKE FOR NOISELESS SYNTHETIC DATA
TABLE II
PERCENTAGE OF SUCCESS RATE OF OUR SNAKE FOR NOISY SYNTHETIC DATA
C. Medical Data
= 05 dB. (a)
= 1:001%. (a)
Fig. 9. Segmentation results for noisy synthetic data with SNR
:
. (b) Barely rejected with J
Barely accepted with J
Rejected with J
:
.
= 0 853%
= 81 065%
In the second experiment, we investigated the sensitivity to
noise of our snake depending on the number of snake coefficients . We generated 100 noisy realizations of a circle with
a radius of 50 pixel units for different SNRs. We computed the
power of the noise over a region of interest of size 200 200.
We illustrate a realization of the resulting images in the header
of Table II.
We show the percentage of success in Table II. We considered
that our snake succeeded in segmenting the circle when the op. This
timization process led to a segmentation with
criterion is very conservative, as shown in Fig. 9. We observe
from the results that our snake is robust against noise since it
is capable of giving a proper segmentation even for low SNRs.
Furthermore, the increased sensitivity to noise as we increase
corresponds to the appearthe number of vector coefficients
ance of additional noise-related local minima in the energy of
the snake. Therefore, should be chosen as small as possible in
order to avoid overfitting of the noise but should be big enough
to be able to approximate the shape of interest.
Now, we move away from synthetic data. We compare our
snake against other snake variants in terms of accuracy and
speed. We quantify their accuracy at outlining the endocardial
wall of the left ventricle within slices of 3-D cardiac MR image
sequences.
The data we used are short-axis cardiac MR image sequences
from 33 subjects acquired in the Department of Imaging of the
Hospital for Sick Children in Toronto, Canada [37]. For each
subject, data consist of a time series of 20 volumes. For each
volume, the number of slices varies from 8 to 15. Each slice
is a 256 256 image with a pixel spacing between 0.93 and
1.64 mm. The ground truth was obtained by manual annotation.
In each segmented image, 1000 points (called landmark points)
define a closed polygon outlining the endocardial wall.
1) Accuracy: For each subject, we selected one slice guided
by its anatomical structures along the long axis and its timing
in the cardiac cycle. Since the region of interest is nearly elliptical, we used the minimalistic elliptical active contour named
Ovuscule to provide a first estimate of the location and the orientation of the left ventricle [28]. Then, we refined the segmentation of the endocardial wall using the general parametric acand several
tive contour model (1) for different values of
basis functions. More specifically, we used linear and quadratic
B-splines, our function (8) that we refer to as a third-order exponential spline, and an extended version of (8) that we refer to as a
fourth-order exponential spline. The linear B-spline basis function has a smaller support than our function (8). However, it can
only adopt the form of polygons. The quadratic B-spline basis
function has the same support and regularity than (8). However,
DELGADO-GONZALO et al.: SNAKES WITH ELLIPSE-REPRODUCING PROPERTY
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Fig. 10. Mean and variance of the landmark error across all 33 patients.
Fig. 11. Median of the landmark error across all 33 patients.
it is unable to reproduce ellipses. Finally, the fourth-order exponential spline is an extended version of (8), with one more
degree of regularity but with a support of one unit larger. The
initialization provided by the Ovuscule could be carried over to
(8) and to the fourth-order exponential spline. In the case of the
other types of snake, the perfect ellipse of the Ovuscule cannot
be reproduced but must be approximated. This approximation
was achieved by sampling the outline of the Ovuscule.
In a preprocessing step, the images were magnified four times
horizontally and vertically. First, we evolved the Ovuscule on
the magnified image. Second, we evolved more refined that were
guided exclusively by the edge energy on a smoothed version
of the magnified image. The smoothing was Gaussian, with a
. We then measured the landmark
kernel of variance
error. We computed this error as the mean distance of the snake
to the landmark points given by the ground truth, as was done
in [37].
In Figs. 10 –12, we show the mean, median, and maximum
values of the landmark error, respectively. From these graphs,
we validate that the Ovuscule provides a good and robust
starting point to be refined by the snakes investigated in this
paper. The polygonal snake does not reach the accuracy of the
and exhibits a high variance across
Ovuscule until
subject. The quadratic-spline snake and the third-order exponential-spline snake converge to similar accuracy starting with
. This was expected since we showed in Section III-B
that our function does converge to a quadratic B-spline when
increases. However, for low values of
, the difference
is noticeable, and the quadratic-spline snakes produce shapes
that are not compatible with the region of interest. Finally,
Fig. 12. Maximum landmark error among all 33 patients.
the fourth-order exponential-spline snakes produce equivalent
results in terms of accuracy and stability than the third-order
exponential-spline snake, at a price of a larger support, and
therefore, of a slower convergence.
In Fig. 13(b), we illustrate the initialization provided to
the Ovuscule and, in Fig. 13(c), the outcome of optimizing
the Ovuscule, which will provide the initialization for further
processing. We also show the result of several more elaborated
snake variants and how they compare with the ground truth.
The fourth-order exponential-spline snake results in an outline
that is visually indistinguishable from that of the third-order
one but comes at an increased computational cost.
2) Speed: In terms of speed, we compared our proposed
snake to some classic traditional snakes such as a Kass-like
snake [38] and a traditional geodesic active contour (GAC)
model [13].
In this analysis, we used the anatomical structures of the 33
patients found in Section V-C1. However, we modified our initialization procedure to accommodate for the GAC model since
it fails unless the initial contour lies totally inside or outside
of the boundary of interest. Therefore, we scaled down the initialization that was provided by the outcome of optimizing the
Ovuscule in Section V-C1. By doing so, we guarantee that all
initial contours lay inside the endocardial wall to segment. Unfortunately, neither the Kass-like snake nor the GAC model are
able to reproduce the initial ellipse perfectly, and their initialization must be approximated. This approximation was achieved
by sampling the outline of the Ovuscule. Finally, we refined the
segmentation of the endocardial wall by using our snake model
for different values of , the Kass-like snake, or the GAC.
This experiment was performed on a MacPro 3.1 with two
Quad-Core Intel Xeon processors and 8 GB of RAM running
Mac OS X 10.6.8. The implementation of the Kass-like active
contour was taken from [38] and one of the GAC model from
the free open-source image-processing package Fiji 1 implementing the algorithm described in [13].
In Fig. 14, we show the mean temporal evolution of the improvement of the Jaccard distance during the snake evolution
process for the 33 patients. We can clearly see that the proposed
snake reaches its optimum earlier than the classical Kass-like
snake and the GAC model. The Kass-like snake has a very costly
first step. Moreover, it cannot escape a local minimum. The
1
http://fiji.sc/
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IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 21, NO. 3, MARCH 2012
M=3
Fig. 14. Temporal evolution of the Jaccard distance. During the 2 s of snake
evolution, the proposed method with
performed 1479 iterations; with
, it performed 1406 iterations; and with
it performed 889
iterations. The Kass snake performed 17 iterations. The first of these iterations
took 370 ms. The GAC performed 34 iterations.
M=5
M=3
D. Real Data
Fig. 13. Outline of the endocardial wall in the first frame and fourth slice of the
second patient. (a) Raw data. (b) Initialization. (c) Ovuscule. (d) Ground truth.
(e) Polygonal snake with
. (f) Quadratic-spline snake with
. (g)
Third-order exponential-spline snake with
. (h) Fourth-order exponen.
tial-spline snake with
M =3
M=4
M=3
M =3
GAC is executed with an advection value of 2.2 and a propagation value of 1. These parameters make the GAC succeed
in overcoming the local minimum, but the convergence rate is
still slower than that of the parametric case. It is important to
notice that, for our proposed model, an increase in the number
of control points slows the convergence. As pointed out in
Section IV-D, this is due to the fact that the larger values of
increase the computational load per iteration of the snake.
Here, we illustrate the behavior of our snake and provide further insights into its capabilities. In the context of this section,
the ground truth is missing; therefore, we must relinquish quantitative assessments in favor of qualitative ones.
1) HeLa Nuclei: We want to evaluate the success of our
snake model at outlining ellipse-like targets in the context of
automated time-lapse microscopy. We use (434 434) images
of HeLa nuclei that express fluorescent core histone 2B on an
RNAi live cell array. We show in Fig. 15 the result of the opti. This number of points is
mization process with (8) and
high enough to capture small departures from an elliptic shape.
We initialized every snake as a circle with a radius of 25 pixel
units, as shown in Fig. 15. These initial circles were centered on
the locations given by a maxima detector applied over a version of the image that was smoothed with a Gaussian kernel of
pixel. A total number of 23 maxima were
variance
detected. We then proceed with an inverted version of the original unsmoothed image to optimize the snakes. The optimization process converged in 22 cases. We show in Fig. 15 the result of the outlining process. We observe that our snakes were
successful in most of the cases.
2) Droplets: As a second example, we show the outline of
sprayed and deformed water droplets hitting a surface. The flight
and the impact of the droplet was captured by a high-speed
camera (Photron Fastcam) at a rate of 10 000 images/s. The
shape of the droplet is changing during flight, at impact, and
while bouncing. After cropping, the size of the image was 663
663 pixels.
We analyzed two frames. One was an image taken before the
collision took place; the other was taken after the impact. In both
cases, we initialized the snake as a circle with a position and
size that we chose manually. These initializations are shown in
Fig. 16. In the image prior to the impact, which we show in
was used. We sethe left part of Fig. 16, a snake with
lected a small value for because the droplet is nearly circular.
In the image after the impact, which we show in the right part
of Fig. 16, five control points did not provide enough freedom
to cope with the discontinuity created by the attachment to the
surface. However, the outline was successfully retrieved when
.
slightly increasing the number of nodes to
DELGADO-GONZALO et al.: SNAKES WITH ELLIPSE-REPRODUCING PROPERTY
1269
Fig. 16. Sprayed droplets. (a) Prior to the impact: The initial contour of the
snake is is represented with a black dashed line. (b) After the impact: The initial
contour of the snake it is represented with a black dashed line. (c) Prior to the
is represented with a white dashed
impact: The outline of our snake with
line. (d) After the impact: The outline of the successful snake it is represented
, while the configuration with
is
with a white dashed line
represented with a gray solid line. The droplet edges are partially out of focus,
making them blurry and noisy.
(M = 8)
M =5
Fig. 15. Outline of HeLa nuclei in a fluorescence microscopy image. The para. (a) Initial contour of the snake. (b) Result
metric snakes were built with
provided by our snake.
The method described in this paper has been programmed as
a plugin for ImageJ, which is a free open-source multiplatform
Java image-processing software.2 Our plugin3 is independent
of any imaging hardware, and due to ImageJ, any common file
format may be used.
VI. CONCLUSION
Our contribution in this paper is a new family of basis
functions that we have used to describe parametric contours in
terms of a set of control points. We were able to single out the
basis of shortest support that allows one to reproduce circles
and ellipses. Those can be characterized exactly by as few as
three control points, but by considering additional ones, our
parametric contours can reproduce with arbitrary precision
any planar closed curve. In particular, we have shown that the
mean error of approximation decays in the inverse proportion
of the cube of the number of control points. We have used our
ellipse-reproducing parametric curves to build snakes driven
2
http://rsb.info.nih.gov/ij/
3
http://bigwww.epfl.ch/algorithms/esnake/
M =5
M=5
by a combination of contour-based and region-based energies.
In the latter case, the energy depends on the contrast between
two regions: one being delineated by the curve itself and the
other by an ellipse of double area. To determine this ellipse,
we showed how to compute the best elliptical approximation,
in a least-squares sense, of a contour described by an arbitrary
number of control points. We were able to accelerate the
implementation of our snakes by taking advantage of Green’s
theorem, which was facilitated by the availability of the explicit
expressions of our basis. We have applied our snakes to a variety
of problems that involve synthetic simulations and real data.
We achieved excellent objective and subjective performance.
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Ricard Delgado-Gonzalo was born in Barcelona,
Spain, in 1983. He received the Diploma in
telecommunications engineering and the Diploma
in mathematics from the Universitat Politècnica
de Catalunya, Barcelona, in 2006 and 2007, respectively. He is currently working toward the
Ph.D. degree with the Biomedical Imaging Group,
École polytechnique fédérale de Lausanne (EPFL),
Lausanne, Switzerland.
In 2008, he joined the Biomedical Imaging Group,
EPFL, as a Ph.D. student and as an Assistant. He is
currently working on applied problems related to image reconstruction, segmentation, and tracking, as well as on the mathematical foundations of signal
processing and spline theory. His research interests include technology management and transfer.
Philippe Thévenaz was born in Lausanne, Switzerland, in 1962. He received the Diploma in microengineering from the École polytechnique fédérale
de Lausanne (EPFL), Lausanne, Switzerland, in
1986 and the Ph.D. degree with a thesis on the use
of the linear prediction residue for text-independent
speaker recognition.
Then, he joined the Institute of Microtechnology,
University of Neuchâtel, Neuchâtel, Switzerland,
where he worked in the domain of image processing
(optical flow) and speech processing (speech coding
and speaker recognition). Then, he worked as a Visiting Fellow with the
Biomedical Engineering and Instrumentation Program, National Institutes of
Health, Bethesda, MD. Since 1998, he has been with the EPFL as a Senior
Researcher. His research interests include splines and multiresolution signal
representations, geometric image transformations, and biomedical image
registration.
Chandra Sekhar Seelamantula (M’99) was born
in Gollepalem, Andhra Pradesh, India, in 1976.
He received the B.E. degree (with honors), with
specialization in electronics and communication
engineering, from the University College of Engineering, Osmania University, Hyderabad, India, in
1999 and the Ph.D. degree from the Indian Institute
of Science, Bangalore, India, in 2005, with a thesis
entitled Time-Varying Signal Models: Envelope and
Frequency Estimation with Applications to Speech
and Music Signal Compression.
During his doctoral years, he specialized in the development of auditorymotivated signal processing models for speech and audio applications. From
April 2005 to March 2006, he worked as a Technology Consultant for M/s. ESQUBE Communication Solutions Pvt. Ltd., Bangalore, and developed proprietary audio coding solutions. In April 2006, he joined the Biomedical Imaging
Group, École polytechnique fédérale de Lausanne, Lausanne, Switzerland, as
a Postdoctoral Fellow and specialized in the field of image processing, optical
coherence tomography, holography, splines, and sampling theories. Since July
2009, he has been an Assistant Professor with the Department of Electrical Engineering, Indian Institute of Science.
DELGADO-GONZALO et al.: SNAKES WITH ELLIPSE-REPRODUCING PROPERTY
Michael Unser (M’89–SM’94–F’99) received the
M.S. (summa cum laude) and Ph.D. degrees in
electrical engineering from the École Polytechnique
Fédérale de Lausanne, Lausanne, Switzerland, in
1981 and 1984, respectively.
From 1985 to 1997, he worked as a Scientist with
the National Institutes of Health, Bethesda, MD. He
is currently a Full Professor and a Director of the
Biomedical Imaging Group, EPFL. He is the author
of 200 journal papers and is one of ISI’s Highly Cited
authors in Engineering ( http://isihighlycited.com).
His main research interest include biomedical image processing. His other
main interests include sampling theories, multiresolution algorithms, wavelets,
and the use of splines for image processing.
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Dr. Unser served as an Associate Editor-in-Chief for IEEE TRANSACTIONS
MEDICAL IMAGING in 2003–2005 and as an Associate Editor for the same
journal in 1999–2002 and in 2006–2007. He also served as an Associate Editor
for the IEEE TRANSACTIONS ON IMAGE PROCESSING in 1992–1995 and for the
IEEE SIGNAL PROCESSING LETTERS in 1994–1998. He is currently a member
of the editorial boards of Foundations and Trends in Signal Processing, and
Sampling Theory in Signal and Image Processing. He coorganized the first IEEE
International Symposium on Biomedical Imaging and was the Founding Chair
of the technical committee of the IEEE-SP Society on Bio-Imaging and Signal
Processing. He was the recipient of the 1995 and 2003 Best Paper Awards, the
2000 Magazine Award, and two IEEE Technical Achievement Awards (2008
SPS and 2010 EMBS). He is an EURASIP Fellow and a member of the Swiss
Academy of Engineering Sciences.
ON